Paper · Dec 3, 2024

Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

Likhit Ganedi, Alice Marveggio, Kerrek Stinson

math.APWider flowsarXiv:2412.02567
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Abstract

We consider a variational model for heterogeneous phase separation, based on a diffuse interface energy with moving wells. Our main result identifies the asymptotic behavior of the first variation of the phase field energies as the width of the diffuse interface vanishes. This convergence result allows us to deduce a Gibbs-Thomson relation for heterogeneous surface tensions. Proceeding from this information, we prove that (weak) solutions of the Allen-Cahn equation with space dependent potential converge to a BV solution of weighted mean curvature flow, under an energy convergence hypothesis. Additionally, relying on the relative energy technique, we establish a weak-strong uniqueness principle for solutions of weighted mean curvature flow.

Mean curvature flowWeak & singular flowsStability & uniquenessPhase fields & multiphase flows

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere (Jingwen Chen and Pedro Gaspar, 2021)8Chen 2021 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)4Hensel 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)6Hensel 2021 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)2Fischer 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)1Laux 2022 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)5Steinke 2023 Stability of multiphase mean curvature flow beyond circular topology changes (Julian Fischer, Sebastian Hensel, Alice Marveggio et al., 2024)3Fischer 2024 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)7Laux 2026
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  1. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Kerrek Stinson · Shared terms “diffuse interface”, “weak strong”, “Allen Cahn”

  2. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

    Julian Fischer and Alice MarveggioWider flows PDF

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same author Alice Marveggio · Shared terms “Allen Cahn”, “Cahn equation”, “interface”

  3. Stability of multiphase mean curvature flow beyond circular topology changes

    Julian Fischer, Sebastian Hensel, Alice Marveggio et al.Wider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Alice Marveggio · Shared terms “uniqueness principle”, “weak strong”, “interface”

  4. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “Allen Cahn”, “Cahn equation”, “weak strong”

  5. Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow

    Pascal SteinkeWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Cahn equation”, “Allen Cahn”, “weak strong”

  6. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “Allen Cahn”, “weak strong”, “strong uniqueness”

  7. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Allen Cahn”, “Cahn equation”, “interface”

  8. Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “Allen Cahn”, “Cahn equation”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.